Question

find the standard form for the equation of a circle $(x - h)^{2}+(y - k)^{2}=r^{2}$ with a diameter that has endpoints $(-3,5)$ and $(1,7)$. $h =$ $k =$ $r =$

Explanation:

Step1: Find the center \((h,k)\)

The center of the circle is the mid - point of the diameter. The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For the points \((-3,5)\) and \((1,7)\), \(h=\frac{-3 + 1}{2}=\frac{-2}{2}=-1\), \(k=\frac{5+7}{2}=\frac{12}{2} = 6\).

Step2: Find the radius \(r\)

The radius is half of the length of the diameter. The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
The length of the diameter \(d=\sqrt{(1-(-3))^2+(7 - 5)^2}=\sqrt{(4)^2+(2)^2}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\).
So \(r=\frac{d}{2}=\sqrt{5}\).

Answer:

\(h=-1\)
\(k = 6\)
\(r=\sqrt{5}\)